The Dynamics of Pendula: An Introduction to Hamiltonian Systems and Chaos Adrianne Stroup [email protected] California Institute of Technology October 1, 2004 Abstract The Laser Interferometer Gravitational Wave Observatory (LIGO) was built to detect and observe gravity waves, whose existence was predicted by Einstein’s theory of general relativity. A critical part of the interferometer are the mirrors which reflect the laser beams, and whose motion determines if a gravity wave is present. As of 2001, LIGO was slated to update their facility by housing each mirror on the lowest bob of a quadruple pendulum to reduce thermal noise. With this system as motivation, I examine simple and multiple-component pendula and use them to convey basic techniques to study Hamiltonian sys- tems. New terminologyand concepts will be introduced when relevant to a perticular pendulum system. By increasing the complexity of the system, the material covered will increase in difficulty. Some of the concepts covered will include: stability, Runge-Kutta integration, nonautonomous systems, Euler- Lagrange Equations, and Poincar´e sections. 1 1 Introduction Nonlinear dynamics is the study of time-evolving systems governed by equa- tions where superposition fails [13]. Numerous analytical and numerical tech- niques have been developed to analyze such dynamical systems. Even elementary systems introduced in one’s first physics class are nonlinear in nature. For example, consider the unforced simple pendulum described by the following differential equation: g θ¨ + sin θ =0, (1) L where g is the constant gravitational acceleration, L is the length of the pendu- lum, and θ is the displaced angle. One uses a small angle approximation (sin x ∼ x + O(x3) for small x) to approximate the motion of the simple pendulum as linear. This simplification eliminates several solutions to a pendulum’s motion such as when a pendulum whirls completely around its pivot [13]. The exact analytical solution of (1) can be expressed in terms of Jacobian elliptic func- tions [3], but one can gain a considerable amount of insight by instead using qualitative methods. 1.1 Hamiltonian Systems Hamiltonian systems comprise a class of dynamical systems in which some quantity (typically energy) is constant along the system’s trajectories [3]. Such systems take the form H(x, p) = constant for some set of displacements and momenta. The associated dynamical system is given by ∂H x˙ = , ∂p ∂H p˙ = − , (2) ∂x which are termed Hamiltons's Equations.ThequantityHis called the Hamilto- nian. If the conserved quantity is energy, it is given by the sum of the potential and kinetic terms. Hamiltonian systems have an equal number of positions and 2 momenta that describe the motion of the system. Each position-momentum pair comprises a degree of freedom (dof). The Hamiltonian structure of a system can be exploited when studying dynamics [16]. Undamped pendula are canonical examples of Hamiltonian systems. The dy- namics of multiple-component pendula are of great practical interest, as evinced by GEO 600, a gravity wave detection facility in Germany that uses triple pen- dula to reduce thermal noise in their interferometers [12]. Additionally the Laser Interferometer Gravitational Wave Observatory (LIGO) was built to detect and observe gravity waves, whose existence was predicted by Einstein’s theory of general relativity. A critical part of the interferometer are the mirrors which re- flect the laser beams, and whose motion determines if a gravity wave is present. As of 2001, LIGO was slated to update their facility by housing each mirror on the lowest bob of a quadruple pendulum to reduce thermal noise [11]. These developments have produced scientific interest in the dynamics of triple pendula [15, 4, 14]. L1 1 m1 L2 2 m2 L3 3 m3 Figure 1: Triple pendulum with mass (m1,m2,m3), length (L1,L2,L3), and angle (θ1,θ2,θ3) labeled. To be described as chaotic, a system must satisfy three general criteria [13]: the motion of the system must be extremely (i.e., exponentially) sensitive to the 3 initial conditions, as small input errors can produce drastically different outputs. It is impossible to predict the motion of a chaotic system after a short period of time because of such sensitivity. Second, chaotic systems are recurrent, in that if one defines a neighborhood (no matter how small) around an initial point in phase space, the trajectory starting at that point will intersect that neigh- borhood infinitely many times during its time-evolution. Third, trajectories do not settle down. As t goes to ∞ the trajectory does not become a fixed point, periodic orbit, or a quasi-periodic orbit [13]. 2 Simple Pendulum We begin our discussion with some terminology from nonlinear dynamics. In the autonomous systemx ˙ = f(x), the phase space is the space with coordinates (x1, ..., xn), which is is n-dimensional [13]. (Autonomous means the system is not dependent on time). The trajectories are solutions of the system plotted in the phase space starting at an initial condition x(0). Along them, the Hamilto- nian is constant. Equilibria occur at points in phase space that satisfy f =0, they are called equilibria because if a trajectory begins at an equilibrium it will remain there for all time. A plot of all the qualitatively different trajectories is called a phase portrait [13]. Equilibria determine the appearance of the phase portrait. The simple pendulum has one dof. With Newtons laws, one can derive the equations of motion for the simple pendulum (1). 2.1 Linearization about the Equilibria Linearization determines the local stability of the equilibrium by computing if a small disturbance from the equilibrium grows or decays [13]. This is done by computing the Jacobian matrix for the system at the specified equilibrium. For the simple pendulum, we begin by writing equation (1) as a pair of first order differential equations 4 θ˙ = v, v˙ = − sin θ. (3) The Jacobian for the simple pendulum is 0 1 01 @ A . (4) −cos θ 0 With the Jacobian, a new matrix A can be computed for each of the equilibria. The equilibria for the simple pendulum are (0, 0) and (π, 0). The matrix A is the Jacobian matrix solved at one of the equilibria. So at (0, 0), A is 0 1 01 @ A. (5) −10 From this matrix we can compute the eigenvalues using the characteristic equa- tion det(A − γI) = 0, where I is the identity matrix. (Solving for γ will yeild √ the eigenvalues). At (0, 0), γ = −1 so the eigenvalues are complex. While at (π, 0), γ =1,−1. (Computation of A at (π, 0) is left for the readers). Then, af- ter solving for the eigenvalues of this matrix the local stability of the equillibria can be determined depending on the values of the eigenvalues. 2.2 Stability of Equilibria The two equilibria of the simple pendulum occur at the apex and the lowest point of the pendulum’s motion. In the previous section, we found the eigen- values at the origin (the lowest point of the pendulum’s swing) to be complex. Complex eigenvalues occur when the equilibrium is a center or a spiral. The distinction is made when the eigenvalues are pure imaginary, when the trace is equal to zero. (The trace is the addition of the diagonal entries of the matrix). Otherwise the equilibria is a spiral. The bottom position of a simple pendulum is a center, for which trajectories nearby are neither attracted nor repelled and remain in periodic orbits nearby. An equilibrium that is a center is stable. To 5 find the direction of the flow compute a few vectors in the vector field [13]. The rotation about the equilibria located at the origin (a center) is clockwise. The pendulum has another equilibrium at θ = π (the top of its motion). The eigenvalues at this equilibrium are positive and negative. This equilibrium is a saddle, so trajectories are both attracted and repelled. Saddles are unstable because of this. With this information and the energy equation (6) of this system, 1 θ˙2 − cos θ = E, (6) 2 where θ˙ is the velocity, and E = constant, one obtains a complete qualitative description of the system’s dynamics (see Figure 2). 2.3 Numerical Analysis of the Simple Pendulum Numerical integration must be used to complement analytical work because differential equations can almost never be exactly solved analytically. Using the Runge-Kutta method [7], we can simulate the single pendulum using Matlab (Figure 2). The phase space for the simple pendulum is cylindrical because θ is a periodic variable. The different saddle points in the phase plane represent the same state, as indicated by the cylindrical phase space. They are all unstable equilibrium at the apex of the pendulum’s motion. (When the pendulum is positioned straight up, any disturbance will cause the pendulum to be repelled from the equilibrium and swing from the apex). The trajectory which intersects the saddle forms a separatrix; on either side of this trajectory the motions are qualitatively different. Trajectories outside the separatrix correspond to periodic whirling motion, while trajectories inside of the separatrix are librations. Librations are small oscillations about an equilib- rium. This is the motion commonly identified with a pendulum: the oscillations about the origin. In a cylindrical phase space, the separatrix becomes a homoclinic orbit where 6 the trajectory begins and ends at the same point in phase space. Homoclinic orbits are common in Hamiltonian systems, but rare in non-conservative sys- tems. (The separatrix would be a pair of heteroclinic orbits if the phase space was planar, two trajectories beginning and ending at equilibria). 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −4 −3 −2 −1 0 1 2 3 4 Figure 2: Phase portrait of the simple pendulum: a plot in the plane (θ, θ˙)where θis from (−π, π].